3. The Mandelbrot Set and Julia Sets

3. C. The Mandelbrot Set

The Mandelbrot set is defined by the same iteration used to define Julia sets, but applied in a different fashion.
The analog of the Mandelbrot set can be defined for any zn + c, for any integer n > 2.
Here is an illustration of the effect of the maximum number of iterations on drawing the Mandelbrot set.
One of the early surprises of the Mandelbrot set is that its periphery is filled with a halo of tiny copies of the entire set, each of which is surrounded by its own halo of still tinier copies, and so on, on smaller and smaller scales, without end.
Despite appearances, these small copies are attached to the main body of the set, through a sequence of still smaller copies. That is, the Mandelbrot set is connected.
The large filled-in components of the Mandelbrot set correspond to stable cycles.
Restricting our attention to just real numbers, the Mandelbrot iteration scheme reveals an interesting relation with the familiar logistic map bifurcation diagram.
Because they are determined by iterating the same function, it is no surprise that the Mandelbrot set and the Julia sets are related.
Finally, here is a gallery of Mandelbrot set images from several websites.

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